0.9999.... = 1.0

[Jay520]

Jay520 said:

MDMAlliance said:

You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

So you have a problem with the decimal system in general then? What's your opinion of 1/9. Writting in decimal form, it is written as 0.111111 (repeating). Do you think this is flawed representation of an actual number? If that's the case, why can't you accept that 0.9999999 (repeating) is a flawed representation of an actual number? That number would be 9 * 0.11111111 (1/9) which is 9/9 which is 1. What's wrong with that logic? 

In fact, the decimal system is only flawed if you assume 0.999999 doesn't equal one. If you accept that it does, then you are free to write fractions like 1/9, 3/9, etc. in decimal form and they would be perfectly valid.

And speaking of series. 0.99999 (repeating) can be expressed as an infinite series. It can be expressed as 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4.....which can be expressed as a sum of an infinite geometric sequence:

 9(1/10)
1-(1/10)

which equals 1

Better picture: